Rollup merge of #135501 - tgross35:stdlib-dependencies-private, r=bjorn3

Inject `compiler_builtins` during postprocessing and ensure it is made private

Follow up of rust-lang/rust#135278

Do the following:

* Inject `compiler_builtins` during postprocessing, rather than injecting `extern crate compiler_builtins as _` into the AST
* Do not make dependencies of `std` private by default (this was added in #135278)
* Make sure sysroot crates correctly mark their dependencies private/public
* Ensure that marking a dependency private makes its dependents private by default as well, unless otherwise specified
* Do the `compiler_builtins` update that has been blocked on this

There is more detail in the commit messages. This includes the changes I was working on in rust-lang/rust#136226.

try-job: test-various
try-job: x86_64-msvc-1
try-job: x86_64-msvc-2
try-job: i686-mingw-1
try-job: i686-mingw-2

Author: matthiaskrgr